Varna.info : Travel to Varna :

Constanta to Varna

You can choose the modes of transport mentioned below to travel from Constanta, Romania to Varna, Bulgaria. The distance between these two cities is 76 miles, which is about 122 km.

Flights Bucharest to Varna
There are no direct flights between Constanta and Varna, but many airlines operate their flights between Bucharest, the capital of Romania, and Sofia, the capital of Bulgaria. The flight time is approximately one hour if there is no stopover. The cost of a one-way trip is around 100 - 200 Euros depending on the airlines.

Travel By Train
There are no direct trains from Constanta to Varna. You have to travel from Constanta to Bucharest, from Bucharest to Ruse and then from Ruse to Varna. The train journey takes around 14 hours including a number of stops along the way.

Travel By Minibus
There are problems crossing the border while using others modes of transportation like regular buses and trains. So people prefer to travel to Varna by minibus. The trip takes just 3 ½ hours. It is safer to travel in groups while using this mode of transportation. During the summer season every day there will be a bus in the route Constanta – Durankulak- Balchik- Kranevo-Albena-Golden Sands- Sunny Day- Varna. The price for one-way ticket is 25 euro.
http://durankulak.net/sitebg/transport.html

Travel By Ferry
There is a seasonal ferry service in summer between Constanta and Varna that starts from Odessa, Ukraine. The trip takes around 3 to 4 hours and costs around 100 Euros. Tourists can catch the ferry Odessa – Constanta – Varna every Friday from 1pm.
http://www.spu.ru/strany/bolgariy/info/v_bolgariy_morem/

Note:
Crossing the border between Romania and Bulgaria is sometimes not a very smooth affair. There are delays due to customs checks and other formalities. It is always good to check with the authorities based on the type of travel documents that you already possess and what you would further need to travel across the border. It is better to be well equipped than to be stranded in an unknown country.